(d)
We could have used the method of substitution here.
(e)
Factor the difference of squares.
Factor the difference of squares again.
NOW TRY
= 1 x^2 + 921 x+ 321 x- 32
= 1 x^2 + 921 x^2 - 92
x^4 - 81
= 19 k+a+ 2219 k- a- 22
81 k^2 - 1 a+ 222 = 19 k 22 - 1 a+ 222 = 19 k+a+ 2219 k- 1 a+ 222
x^2 - y^2 = 1 x + y 2 1 x - y 2
334 CHAPTER 6 Factoring
⎩⎨⎧ ⎩⎨⎧
CAUTION Assuming that the greatest common factor is 1, it is not possible
to factor (with real numbers) a sum of squaressuch as in Example 1(e).In
particular, x^2 +y^2 Z 1 x+y 22 ,as shown next.
x^2 + 9
OBJECTIVE 2 Factor a perfect square trinomial. Two other special products
from Section 5.4lead to the following rules for factoring.
Perfect Square Trinomial
x^2 2 xyy^2 1 xy 22
x^2 2 xyy^2 1 xy 22
Because the trinomial is the square of it is called a perfect
square trinomial.In this pattern, both the first and the last terms of the trinomial
must be perfect squares. In the factored form twice the product of the first
and the last terms must give the middle term of the trinomial.
Perfect square trinomial; Not a perfect square trinomial;
middle term would have to be
and 212 m 2152 = 20 m. 16 por - 16 p.
4 m^2 = 12 m 22 , 25= 52 ,
4 m^2 + 20 m+ 25 p^2 - 8 p+ 64
1 x+y 22 ,
x^2 + 2 xy+y^2 x+ y,
NOW TRY
EXERCISE 1
Factor each polynomial.
(a)
(b)
(c)
(d)v^4 - 1
1 a+b 22 - 25
9 x^2 - 729
4 m^2 - 25 n^2
NOW TRY ANSWERS
- (a)
(b)
(c)
(d) 1 v 2 + 121 v+ 121 v- 12
1 a+b+ 521 a+b- 52
91 x+ 921 x- 92
12 m+ 5 n 212 m- 5 n 2
Factoring Perfect Square Trinomials